Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1365 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,12,13,29)(7,18)(8,23)(9,21)(10,26)(14,17,19,16)(15,32,25,34)(20,27)(22,35)(24,33)(28,36,31,30);(5,6,8,14,20,31,33,36,10,17,24,29,9,25,23,34,11,13,7,19,21,28,35,30,26,16,22,12,27,15,18,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,e^2=fgi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216259, #72216264, #72216267, … |
| maximal normal subgroup of | #72216259, #72216264, #72216267, … |
| maximal quotient of | #72225456, #72225489, #72227044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1560);